Mellin motives, periods and high energy physics

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The proposed research lies at the intersection of theoretical mathematics and particle physics. It involves the study of a particular class of numbers called periods, of which pi is the most famous example. Periods are some of the most frequently occurring quantities in mathematics. Their name comes from their origin in the 17th century, when Galilei studied the amount of time it takes a pendulum to complete a swing, and Kepler asked the same question about a planet travelling around the Sun. In modern mathematics, periods are defined much more generally; they play an essential role in many central questions in the classical fields of geometry and number theory. In the proposed research I will work to uncover and study new structures which periods satisfy. Particle physics comes into play via another class of numbers, called Feynman integrals, which are laboriously computed by physicists in order to make predictions for particle collider experiments such as the ones performed in the Large Hadron Collider at CERN. These experiments are necessary for validating existing particle physics theories or finding illuminating discrepancies in them, and have previously led to the discovery of new elementary particles such as the Higgs boson. The difficulty in computing Feynman integrals thus presents a major obstacle for advancements in the field. Fortunately, Feynman integrals are examples of periods, and this project involves applying tools from algebraic geometry to answer questions about their structure, in particular the structure captured by an object called the Cosmic Galois group. This work will help physicists compute Feynman integrals, expanding the scope of physical processes which are possible to predict via quantum field theory. In addition, patterns encountered by physicists can help uncover structures that hold for periods in general, which, in turn, can be used to solve problems in pure mathematics. It is this interplay of mathematics and physics that makes the proposed research during the fellowship particularly exciting.